{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:30:58Z","timestamp":1760059858781,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T00:00:00Z","timestamp":1752537600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Brazilian agencies Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico (CNPq)","award":["04577\/2021-0","307512\/2023-1","302414\/2022-3","308840\/2023-2"],"award-info":[{"award-number":["04577\/2021-0","307512\/2023-1","302414\/2022-3","308840\/2023-2"]}]},{"name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior (CAPES)","award":["04577\/2021-0","307512\/2023-1","302414\/2022-3","308840\/2023-2"],"award-info":[{"award-number":["04577\/2021-0","307512\/2023-1","302414\/2022-3","308840\/2023-2"]}]},{"name":"Funda\u00e7\u00e3o de Amparo a Ci\u00eancia e Tecnologia do Estado de Pernambuco (FACEPE)","award":["04577\/2021-0","307512\/2023-1","302414\/2022-3","308840\/2023-2"],"award-info":[{"award-number":["04577\/2021-0","307512\/2023-1","302414\/2022-3","308840\/2023-2"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We revisit the problem of random search walks in the two-dimensional (2D) space between concentric absorbing annuli, in which a searcher performs random steps until finding either the inner or the outer ring. By considering step lengths drawn from a power-law distribution, we obtain the exact analytical result for the search efficiency \u03b7 in the ballistic limit, as well as an approximate expression for \u03b7 in the regime of searches starting far away from both rings, and the scaling behavior of \u03b7 for very small initial distances to the inner ring. Our numerical results show good overall agreement with the theoretical findings. We also analyze numerically the absorbing probabilities related to the encounter of the inner and outer rings and the associated Shannon entropy. The power-law exponent marking the crossing of such probabilities (equiprobability) and the maximum entropy condition grows logarithmically with the starting distance. Random search walks inside absorbing annuli are relevant, since they represent a mean-field approach to conventional random searches in 2D, which is still an open problem with important applications in various fields.<\/jats:p>","DOI":"10.3390\/e27070758","type":"journal-article","created":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T15:01:05Z","timestamp":1752591665000},"page":"758","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Random Search Walks Inside Absorbing Annuli"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-5842-548X","authenticated-orcid":false,"given":"Anderson S.","family":"Bibiano-Filho","sequence":"first","affiliation":[{"name":"Laborat\u00f3rio de F\u00edsica Te\u00f3rica e Computacional, Departamento de F\u00edsica, Universidade Federal de Pernambuco, Recife 50670-901, PE, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-4006-5569","authenticated-orcid":false,"given":"Jandson F. O.","family":"de Freitas","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal de Vi\u00e7osa, Vi\u00e7osa 36570-900, MG, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3865-2621","authenticated-orcid":false,"given":"Marcos G. E.","family":"da Luz","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal do Paran\u00e1, Curitiba 81531-980, PR, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2301-5593","authenticated-orcid":false,"given":"Gandhimohan M.","family":"Viswanathan","sequence":"additional","affiliation":[{"name":"Department of Physics and National Institute of Science and Technology of Complex Systems, Federal University of Rio Grande do Norte, Natal 59078-970, RN, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4178-8266","authenticated-orcid":false,"given":"Ernesto P.","family":"Raposo","sequence":"additional","affiliation":[{"name":"Laborat\u00f3rio de F\u00edsica Te\u00f3rica e Computacional, Departamento de F\u00edsica, Universidade Federal de Pernambuco, Recife 50670-901, PE, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Grebenkov, D., Metzler, R., and Oshanin, G. (2024). Target Search Problems, Springer.","DOI":"10.1007\/978-3-031-67802-8"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"430301","DOI":"10.1088\/1751-8121\/42\/43\/430301","article-title":"The Random Search Problem: Trends and Perspectives","volume":"42","author":"Grosberg","year":"2009","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1103\/RevModPhys.87.483","article-title":"L\u00e9vy walks","volume":"87","author":"Zaburdaev","year":"2015","journal-title":"Rev. Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1103\/RevModPhys.83.81","article-title":"Intermittent search strategies","volume":"83","author":"Loverdo","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"M\u00e9ndez, V., Campos, D., and Bartumeus, F. (2014). Stochastic Foundations in Movement Ecology, Springer.","DOI":"10.1007\/978-3-642-39010-4"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Viswanathan, G.M., da Luz, M.G.E., Raposo, E.P., and Stanley, H.E. (2011). The Physics of Foraging, Cambridge University Press.","DOI":"10.1017\/CBO9780511902680"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"015062","DOI":"10.1088\/2632-2153\/adbc46","article-title":"Stochastic Resetting Mitigates Latent Gradient Bias of SGD from Label Noise","volume":"6","author":"Bae","year":"2025","journal-title":"Mach. Learn. Sci. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"100559","DOI":"10.1016\/j.cosrev.2023.100559","article-title":"Advances in nature-inspired metaheuristic optimization for feature selection problem: A comprehensive survey","volume":"49","author":"Nssibi","year":"2023","journal-title":"Comput. Sci. Rev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2965","DOI":"10.1038\/s41467-023-38706-z","article-title":"Plasmodium sporozoite search strategy to locate hotspots of blood vessel invasion","volume":"14","author":"Formaglio","year":"2023","journal-title":"Nat. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Brock, O., and Krichmar, J. (2025). From Animals to Animats 17. SAB 2024, Springer. Lecture Notes in Computer Science.","DOI":"10.1007\/978-3-031-71533-4"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1038\/44831","article-title":"Optimizing the success of random searches","volume":"401","author":"Viswanathan","year":"1999","journal-title":"Nature"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"041108","DOI":"10.1103\/PhysRevE.64.041108","article-title":"Average time spent by L\u00e9vy flights and walks on an interval with absorbing boundaries","volume":"64","author":"Buldyrev","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/S0378-4371(01)00461-7","article-title":"Properties of L\u00e9vy flights on an interval with absorbing boundaries","volume":"302","author":"Buldyrev","year":"2001","journal-title":"Physica A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"054147","DOI":"10.1103\/PhysRevE.106.054147","article-title":"L\u00e9vy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal L\u00e9vy walk strategy for random searches","volume":"106","author":"Matos","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"048901","DOI":"10.1103\/PhysRevLett.126.048901","article-title":"Comment on \u201cInverse square L\u00e9vy walks are not optimal search strategies for d \u2a7e 2\u201d","volume":"126","author":"Buldyrev","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"080601","DOI":"10.1103\/PhysRevLett.124.080601","article-title":"Inverse square L\u00e9vy walks are not optimal search strategies for d \u2a7e 2","volume":"124","author":"Levernier","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"048902","DOI":"10.1103\/PhysRevLett.126.048902","article-title":"Reply to \u201cComment on \u2018Inverse square L\u00e9vy walks are not optimal search strategies for d \u2265 2\u2019\u201d","volume":"126","author":"Levernier","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","unstructured":"Zolotarev, V.M., and Uchaikin, V.M. (1999). Chance and Stability, VSP BV."},{"key":"ref_19","unstructured":"Samorodnitsky, G., and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes, Chapman and Hall."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"e2004369118","DOI":"10.1073\/pnas.2004369118","article-title":"Scale-free movement patterns in termites emerge from social interactions and preferential attachments","volume":"118","author":"Paiva","year":"2021","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Bartumeus, F., Raposo, E.P., Viswanathan, G.M., and da Luz, M.G.E. (2014). Stochastic optimal foraging: Tuning intensive and extensive dynamics in random searches. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0106373"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"188103","DOI":"10.1103\/PhysRevLett.109.188103","article-title":"Diffusion of Integral Membrane Proteins in Protein-Rich Membranes","volume":"109","author":"Jeon","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"098303","DOI":"10.1103\/PhysRevLett.116.098303","article-title":"Interfacial molecular searching using forager dynamics","volume":"116","author":"Monserud","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"012126","DOI":"10.1103\/PhysRevE.102.012126","article-title":"Transient dynamics in a nonequilibrium superdiffusive reaction-diffusion process: Nonequilibrium random search as a case study","volume":"102","author":"Rusch","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"118001","DOI":"10.1103\/PhysRevLett.127.118001","article-title":"Anomalous diffusion and L\u00e9vy walks distinguish active from inertial turbulence","volume":"127","author":"Mukherjee","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1038\/nature06948","article-title":"A L\u00e9vy flight for light","volume":"453","author":"Barthelemy","year":"2008","journal-title":"Nature"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2799","DOI":"10.1364\/OL.44.002799","article-title":"L\u00e9vy spectral intensity statistics in a Raman random fiber laser","volume":"44","author":"Li","year":"2019","journal-title":"Opt. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"D155","DOI":"10.1364\/AO.383477","article-title":"Nonlinear effects and photonic phase transitions in Nd3+-doped nanocrystal-based random lasers","volume":"59","author":"Moura","year":"2020","journal-title":"Appl. Opt."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2946","DOI":"10.1103\/PhysRevLett.73.2946","article-title":"Stochastic Process with Ultraslow Convergence to a Gaussian: The Truncated L\u00e9vy Flight","volume":"73","author":"Mantegna","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1103\/PhysRevE.52.1197","article-title":"Analytic approach to the problem of convergence of truncated L\u00e9vy flights towards the Gaussian stochastic process","volume":"52","author":"Koponen","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"010103(R)","DOI":"10.1103\/PhysRevE.100.010103","article-title":"Why L\u00e9vy \u03b1-stable distributions lack general closed-form expressions for arbitrary \u03b1","volume":"100","author":"Rocha","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_32","unstructured":"Abramowitz, M., and Stegun, I.A. (1964). Handbook of Mathematical Functions, Dover."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"021116","DOI":"10.1103\/PhysRevE.76.021116","article-title":"Fractional Laplacian in Bounded Domains","volume":"76","author":"Zoia","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"032113","DOI":"10.1103\/PhysRevE.94.032113","article-title":"L\u00e9vy flights between absorbing boundaries: Revisiting the survival probability and the shift from the exponential to the Sparre-Andersen limit behavior","volume":"94","author":"Raposo","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1080\/01621459.1976.10480344","article-title":"A Method for Simulating Stable Random Variables","volume":"71","author":"Chambers","year":"1976","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"014036","DOI":"10.1088\/0031-8949\/2009\/T135\/014036","article-title":"Simulations of L\u00e9vy flights","volume":"2009","author":"Pantaleo","year":"2009","journal-title":"Phys. Scr."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"034124","DOI":"10.1103\/PhysRevE.106.034124","article-title":"Effect of the search space dimensionality for finding close and faraway targets in random searches","volume":"106","author":"Viswanathan","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Attard, P. (2002). Thermodynamics and Statistical Mechanics: Equilibrium by Entropy Maximisation, Academic Press.","DOI":"10.1016\/B978-012066321-7\/50005-2"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"062135","DOI":"10.1103\/PhysRevE.92.062135","article-title":"Efficient search of multiple types of targets","volume":"92","author":"Wosniack","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"052119","DOI":"10.1103\/PhysRevE.91.052119","article-title":"Robustness of optimal random searches in fragmented environments","volume":"91","author":"Wosniack","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Niebuhr, B.B.S., Santos, M.C., Wosniack, M.E., Raposo, E.P., Viswanathan, G.M., da Luz, M.G.E., and Pie, M.R. (2015). Survival in patchy landscapes: The interplay between dispersal, habitat loss and fragmentation. Sci. Rep., 5.","DOI":"10.1038\/srep11898"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"e09056","DOI":"10.1111\/oik.09056","article-title":"Foraging personalities modify effects of habitat fragmentation on biodiversity","volume":"2022","author":"Jeltsch","year":"2022","journal-title":"Oikos"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"20220103","DOI":"10.1098\/rsif.2022.0103","article-title":"Group size and resource fractality drive multimodal search strategies: A quantitative analysis on group foraging","volume":"19","author":"Nauta","year":"2022","journal-title":"J. R. Soc. Interface"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1093\/beheco\/arac068","article-title":"Patch quality and habitat fragmentation shape the foraging patterns of a specialist folivore","volume":"33","author":"Crowther","year":"2022","journal-title":"Behav. Ecol."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/7\/758\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:10:17Z","timestamp":1760033417000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/7\/758"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,15]]},"references-count":44,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["e27070758"],"URL":"https:\/\/doi.org\/10.3390\/e27070758","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,7,15]]}}}